Ecological Modelling 185 (2005) 105–131
Tropical deforestation in Madagascar: analysis using hierarchical,
spatially explicit, Bayesian regression models
Deepak K. Agarwala,∗ , John A. Silander Jr.b , Alan E. Gelfandc ,
Robert E. Deward , John G. Mickelson Jr.e
a AT&T Shannon Research Labs, Florham Park, NJ 07932-0791, USA
Department of Ecology and Evolutionary Biology, University of Connecticut, Storrs, CT 06269-3043, USA
c Department of Statistics and Decision Sciences, Duke University, Durham, NC 27708-0251, USA
d Department of Anthropology, University of Connecticut, Storrs, CT 062692176, USA
Center for International Earth Science Information Network, Columbia University, Palisades, NY 10964, USA
b
e
Received 4 August 2003; received in revised form 17 August 2004; accepted 29 November 2004
Abstract
Establishing cause–effect relationships for deforestation at various scales has proven difficult even when rates of deforestation
appear well documented. There is a need for better explanatory models, which also provide insight into the process of deforestation. We propose a novel hierarchical modeling specification incorporating spatial association. The hierarchical aspect allows us
to accommodate misalignment between the land-use (response) data layer and explanatory data layers. Spatial structure seems
appropriate due to the inherently spatial nature of land use and data layers explaining land use. Typically, there will be missing
values or holes in the response data. To accommodate this we propose an imputation strategy. We apply our modeling approach
to develop a novel deforestation model for the eastern wet forested zone of Madagascar, a global rain forest “hot spot”. Using
five data layers created for this region, we fit a suitable spatial hierarchical model. Though fitting such models is computationally
much more demanding than fitting more standard models, we show that the resulting interpretation is much richer. Also, we
employ a model choice criterion to argue that our fully Bayesian model performs better than simpler ones. To the best of our
knowledge, this is the first work that applies hierarchical Bayesian modeling techniques to study deforestation processes. We
conclude with a discussion of our findings and an indication of the broader ecological applicability of our modeling style.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Tropical deforestation; Bayesian statistical hierarchical models; Spatially explicit regression models; Misaligned spatial data; Human
population pressure; Land-use classification; Statistical model comparison; Multiple imputation
1. Introduction
∗ Corresponding author. Tel.: +1 9733607154;
fax: +1 9733608178.
E-mail address: [email protected] (D.K. Agarwal).
0304-3800/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2004.11.023
The demise of the world’s tropical rain forests has
been of central concern to conservation biologists for
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at least 20 years (Singh, 2001; Whitmore, 1998). But
cognizance of the problem was apparent decades earlier (cf. Jarosz, 1993; Perrier de la Bâthie, 1921). Yet
it has proven surprisingly difficult and controversial to
obtain accurate estimates of deforestation rates or indeed to relate current forest cover to potential or historical extent of forest cover. This remains true even with
the advent of satellite imagery from the 1970s (Apan,
1999). The difficulties arise from issues of definition
and benchmarks, spatial and temporal resolution, and
data adequacy (Singh, 2001; Angelsen and Kaimowitz,
1999). As a consequence even the most recent estimates of deforestation rates, or more generally landuse change, vary considerably, and debate continues
over whether the rates of deforestation in the tropics
are declining or increasing (FAO, 2001; Matthews et
al., 2000). Only for specific, narrowly defined regions
can one obtain reasonably reliable data on deforestation rates over specific time periods (e.g., Mertens and
Lambin, 2000).
Establishing cause–effect relationships for deforestation or land use at various scales has also proven
difficult even when the process appears well documented (Angelsen and Kaimowitz, 1999; Irwin and
Geoghegan, 2001; Barbier, 2001). The standard explanation for deforestation in the tropics has been rapid
population growth, associated poverty, and consequential environmental destruction (Leach and Mearns,
1988; Richards and Tucker, 1988; Mercier, 1991;
Brown and Pearce, 1994; Sponsel et al., 1996). In selected areas, commercial exploitation and clear cutting are also important causes of deforestation (Torsten,
1992). However, conventional explanations for forest
loss and environmental degradation have recently been
questioned as too simplistic, general, or even misleading (Barbier, 2001). In the absence of a clear understanding of the role that various contributing explanatory variables may play in deforestation processes,
let alone establishing cause effect relationships, it is
not surprising that reforestation or forest conservation schemes have had relatively little or no success.
Most of the reports on failures are buried in the gray
or secondary literature (e.g., Bloom, 1998; IUFRO,
2001; Sharma et al., 1994; cf. Oates, 1999); relatively
few are to be found in the primary literature (e.g.,
Olson, 1984; Elster, 2000); and rarely does one find
reports of successful reforestation either via natural
succession (e.g., Guariguata and Ostertag, 2001) or
by human management practices (e.g., Lamb et al.,
1997).
In this paper, we focus on deforestation processes
and patterns of land use in the eastern wet tropical
forests of Madagascar (Fig. 1). The prevailing perception is that the patterns and processes here are well
understood. In fact the land-use change maps of Green
and Sussman (1990) are commonly reproduced in textbooks as standard examples of deforestation processes.
Nevertheless current and historical patterns of deforestation in Madagascar remain poorly understood,
and controversial (e.g., Jarosz, 1993 versus Green and
Sussman, 1990). Current estimates of forest cover in
Madagascar vary by at least five-fold, from 42,000 km2
for “closed forest,” up to 158,000 km2 for “natural forest,” to 232,000 km2 for “total forest and woodland,”
respectively, about 7, 27 and 40% of the land area
of Madagascar (UNEP, 1998; UNEP-WCMC, 2001).
This underscores the pervasive difficulty of estimating
forest cover, which in part reflects differing definitions
of forest (cf. Silander, 2000). In contrast, the forest
cover in 1902 was estimated at about 20% of the land
area (Pelet, 1902). In 1921, based on extensive field
reconnaissance from 1900 to 1915, Perrier de la Bâthie
estimated that about 19% (110,000 km2 ) of Madagascar was in forest or woodland of one sort or another. But
of this only about 35–70,000 km2 (6–12%) was closed
forest. In 1934, forest cover was estimated at only 10%
of the land area (Grandidier, 1934). These figures conflict with estimates of forest cover in the 1960s: 33%
forested in the early to mid 1960s (Humbert and Cours
Darne, 1965) versus 21% a few years later (Le Bourdiec
et al., 1969). The fact that these contemporary estimates
of forest cover are so different is surprising, given the
availability of aerial photography and more precise cartographic methodology. Satellite data from the 1970s
onward have been able to provide only limited estimates of forest cover or deforestation rates. Virtually
all of the available images of the wet tropical forests of
the eastern Madagascar were too obscured by clouds
through the 1990s to provide adequate estimates of forest cover, especially for the focal region of this study.
In fact, the most recent available maps reproduce “current” forest cover for most of the east coast from the
maps published in 1965 (Faramalala, 1995; Du Puy
and Moat, 1999; see also Green and Sussman, 1990)
with no new data. These sources thus indicate current
forest cover for our focal region (Taomasina Province)
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
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Fig. 1. Color composite AVHRR satellite image of Madagascar. The box encompassing Taornasina Province shows the focal area for the study.
The light and dark green areas in along the east coats of Madagascar correspond roughly to potential forest and mature, continuous forest,
respectively.
at 66%, while the center of our area (Fenerive Prefecture) would be 95% forest cover. Clearly this is a gross
overestimate of actual forest cover (see results below),
and a marked departure from all earlier estimates in the
twentieth century, and is indeed at odds with historical
accounts from the 17th through 19th centuries. Even
Green and Sussman’s seminal 1990 study of deforestation rates in Madagascar are based largely on the same
set of maps provided by Humbert et al. (1964–1965).
The confusion and contradiction described above,
underscores the need for a more critical, in depth ex-
amination of deforestation or more generally land-use
patterns and processes. It also points up the need for
evaluating land-use patterns in a more thorough historiography context. Moreover, if there is any indication
that patterns of land-use change spatially from one region to another, this needs to be addressed or one may
risk obscuring pattern and process across scales.
With these objectives in mind, we present a novel hierarchical Bayesian model for explaining deforestation
processes applied to deforestation of Madagascar. Our
method is able to address several issues that arise in the
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study of deforestation processes, which, to the best of
our knowledge, has not been addressed before. First, we
provide a framework to connect data layers that are spatially misaligned. In our example, we explain land-use
patterns (measured at a resolution of 1 km × 1 km) in
terms of population counts available at a much coarser
resolution (by townships). We show that naive schemes
like allocating population uniformly to all pixels within
a given township perform poorly compared to the multilevel hierarchical Bayesian model we suggest. Next,
we provide a novel multiple imputation method to deal
with missing data when modeling spatial processes like
deforestation. Note that “filling up holes” created due
to missing land-use classification using a crude method
like “majority vote due to neighbors” might fail to provide correct estimates of uncertainty for model parameters. Finally, our model can explicitly incorporate spatial structure, which has hitherto not been addressed in
the context of explanatory models that study deforestation processes. Apart from providing better statistical
inference by modeling excess heterogeneity present in
the data, maps of estimated spatial structure are informative and can often lead to new ecological insights.
However, this comes at a price of being able to fit complex hierarchical Bayesian models, which is a computationally challenging undertaking and may involve
writing special purpose code as is the case here. Hierarchical modeling is just beginning to receive attention in the ecology literature (see our brief review of
this work at the end of Section 2). Beyond the obvious applications to land change problems, hierarchical
modeling has broad application to other ecological patterns and processes which present many of the same issues listed above. For example, in attempting to explain
species presence–absence, abundance or richness using ecological explanatory variables, such as climate,
soil geology or topography, one must address missing data, spatially misaligned explanatory data layers,
and spatial structure in the ecological pattern. Because
we had some a priori indication that there were subregional differences in land use within our study region, we subdivided the analysis spatially into separate
northern and southern sub-regions. Finally we interpret
the results of the model in the context of the critical
historical information available. Because the modeling approach we take is novel and technically complex, we start with detailed discussion of the modeling
approaches.
2. Model background and modeling issues
The goal of this paper is thus to develop, fit, and
interpret stochastic models to clarify spatial processes
that can explain patterns of deforestation or more generally land-use patterns. This is a rather challenging
undertaking on several accounts.
First, there are numerous factors which have been
linked to land use and deforestation. These factors can
be socio-economic, e.g., population growth or economic growth; physical factors, e.g., topography or
proximity of rivers and roads; policy-driven government intervention, e.g., agriculture and/or forestry policies; external factors, e.g., demand for exports or financing conditions. For any given region, typically
only some of this information is available or selectively
available at different spatial scales.
Second, there is no well-accepted notion of a response variable. In some cases deforestation rates or
forest areas may be used. Deforestation rates are typically calculated over limited time spans. Forested areas
are usually obtained from cross-sectional studies of different countries or different regions within countries.
The alternative, which we adopt is to partition the study
region into disjoint areal units and then attach an ordinal land-use classification variable to each unit. This
classification represents successive departure from a
potential, fully forested landscape, which all evidence
points to for eastern Madagascar. Of course, such a
variable is not uniquely defined in terms of number of
and definition of classifications.
Third, the explanatory variables and the response
variables are typically measured in different areal units.
For instance, in the dataset we investigate, the response
variable is land-use classification (e.g., forested, nonforested, etc.), which is ascribed to 1 km × 1 km pixels derived from satellite images. On the other hand,
population is recorded at various administrative areal
units. In our case, we use the equivalent of “township”
level data, which are considered to be the most reliable
census data at the finest spatial resolution. The resultant challenge is to develop a regression model for data
collected on spatially misaligned areal units, i.e., areal
units (pixels, polygons, etc.) that do not correspond to
each other or line up as GIS data layers.
Finally, land use and deforestation are inherently
spatial processes. So too, are many of the explanatory variables, such as population counts. Satisfying
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
stochastic, as well as mechanistic objectives, modeling
should capture association between measurements on
areal units in terms of proximity of these units. One
way to accomplish this is to introduce appropriate sets
of spatial random effects, which can serve as spatial surrogates for unmeasured or unavailable covariates that
are inevitable in any modeling protocol.
Little formal modeling of deforestation was attempted until the 1980s (Granger, 1998). Descriptive
statistical summaries of land use or forest area obtained
for certain regions at certain time points were customary. Most of the ensuing statistical work, which has
appeared has been based on standard multiple linear
regression models relating deforestation rates or forest area to a long list of potential explanatory variables
(Granger, 1998). In fact, Granger lists at least 28 different variables, which have been linked directly or indirectly to deforestation or land-use change in a forested
landscape, while Anglesen and Kaimowitz list 140
different models developed to explain just economic
causes of deforestation. In these studies it appears that
the major objective has been simply to maximize R2 ;
and models with eight or more explanatory variables
have been put forward. This is no longer considered
sufficient or useful in statistical modeling these days.
Moreover, none of these models are explicitly spatial
in nature. The little work with an explicitly spatial flavor, which does exist has arisen from an econometric
perspective introduced by Palloni (1992) and Chomitz
and Gray (1996), and more recently summarized by
Irwin and Geoghegan (2001). These models assume
that land use will be devoted to the activity yielding the
highest “rent”. The modeling connects output prices to
input prices through “production” functions with the
spatial aspect introduced solely by relating these prices
to market distance. These models remain quasi-spatial
and econometric.
Hence, we claim that despite the already substantial literature, there is a need for better, explanatory
models, which may provide better insight into the process of deforestation or land-use change (cf. Irwin and
Geoghegan, 2001; Veldkamp and Lambin, 2001). Various modeling lineages have been developed and applied to explain deforestation with questionable success and little attempt at synthesis. These include deterministic and stochastic models, statistical and simulation models, phenomenological and mechanistic models, spatial and non-spatial models, etc.
109
In general, stochastic models provide considerable
advantage over deterministic approaches in situations
where substantial variability is always present, as in
land-use change data. Stochastic modeling allows one
to resolve the signal and to clarify the nature of the
noise, which obscures it. The development of satisfying stochastic models does raise a number of methodological issues. Here, we offer an overview of a critical
subset of these issues. We address the specific details
of these issues later as part of the development section.
First, there is a matter of modeling style. Do we
employ simulation modeling, say as is frequently done
with cellular automata models, attempting to capture
mechanistic aspects of the process? Or do we employ
phenomenological modeling to attempt to relate different data layers, each measuring different variables?
Where there is theory to guide us, can we be somewhat
mechanistic in relating data layers? Finally, do we take
a static or dynamic view of the process? There is no
simple right or wrong answer to these choices.
If we seek to explain land use, then we first have
to formally define the response variable. Even within a
specific context there is not usually a well-accepted notion. Also, we have to identify which factors we seek to
link to land-use. Do we seek a socio-economic explanation? Do we seek a physical or ecological explanation?
Here we must recognize that no proposed explanatory
model is correct, but that some are more useful than
others and that no proposed explanatory model establishes causality, merely relationship. Moreover, models employing different subsets of variables can explain
comparably well. One must decide which perspective
to explore and which subset of possible factors to connect.
When data layers are to be interrelated, they can be
introduced at various levels of a hierarchical specification, which becomes an explanatory model for the process (cf. Wu and David, 2002). In this specification the
lowest level is the response. This modeling strategy is
referred to as hierarchical or multilevel modeling. Here,
for example, the hierarchical levels of the full model
include a population model, a land-use classfication
model, and a model specifying latent variables. This approach allows for considerable flexibility in modeling
and the possibility of incorporating mechanistic components in linking certain levels. Hierarchical modeling
is achieving increased utilization in analyzing data collected from complex processes. See the recent papers
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by Wikle (2003), by Clark et al. (2004), and by Gelfand
et al. (2003, 2005).
Full inference for hierarchical models is most easily
achieved (and often only possible) within a Bayesian
framework. In particular, this framework enables a posterior distribution (i.e., a conditional distribution given
the observed data) for all model unknowns. This distribution allows any desired inference about any of the
unknowns. Such models are fitted using simulationbased methods, e.g., Gibbs sampling and Markov chain
Monte-Carlo methods. See Carlin and Louis (2000) and
Gelman et al. (1995) for an introductory presentation
of this methodology.
In the context of explaining land use, the data layers
are inherently spatial. Some are observed at particular
point locations, and some are obtained through satellite
imagery as raster pixel values at some degree of resolution. Other data may be associated with formal governmental units such as towns or other municipal subdivisions, stored as polygons. Data, which is obviously
spatial should be modeled in an explicitly spatial fashion. Only rarely has this actually been done in modeling
land-use patterns, and where spatial models have been
developed, the analysis is not entirely spatially continuous (Irwin and Geoghegan, 2001). Developing spatial
process models that lack spatial contiguity is analogous
to developing models of time series processes that lack
knowledge of the time order of the processes—not very
useful (Irwin and Geoghegan, 2001). While there may
be issues of model choice in providing spatial structure
as well as the scale or resolution at which the modeling
is done, it seems clear that measurements which are
closer to each other in space should be more strongly
associated than those farther from each other. Spatial
dependence should be incorporated, but since different spatial data layers are often collected at different
scales, one needs a formal (rather than ad hoc) modeling strategy to carry out full inference in the presence
of such consequential misalignment. Bayesian hierarchical modeling provides this framework.
In this regard, some spatial features can be described
through the mean structure of the model (or in the case
of categorical or count data, the mean on a transformed
scale). For instance, one can introduce distance to a
road or to a town center with a coefficient as part of a
regression structure. Or one can attempt to explain spatial features through a trend surface in the mean (e.g.,
a polynomial of low order in latitude and longitude).
But there will always be omitted, perhaps unobservable, explanatory variables, which also provide spatial
explanation. In the absence of these variables, spatial
association in the residuals will be expected, and hence
models introducing spatial dependence are needed.
For at least some data layers there will be missing
data. For example, land-use classifications obtained by
remotely sensed satellite data, particularly for wet tropical regions, will have portions obscured (hence missing) by cloud cover. Typically missing data are treated
in an ad hoc manner. Models of spatial association for
gridded data assume there are no missing observations.
To accommodate such holes in the dataset, imputation
(i.e., statistical simulation of missing values) provides
an alternative to subjective assignments of values. That
is to say, one imputes observations at the missing locations according to the likelihood of the different possible values. Such imputation must be done randomly
and, moreover, multiple, hence variable, imputations
are necessary in order to capture the effect of the uncertainty associated with imputation and to better reflect
variability associated with inference in the presence of
missing values.
Lastly, the multilevel hierarchical approach allows
for the specification of many different models; there
is flexibility at each level. But, with many possible
models to consider, tools for model comparison are
required. Simple notions like R2 are clearly inappropriate for multilevel models with categorical or count
variables as response. Criteria, which reflect the utility for the model emerge as more attractive and better
suited. However, such comparison is really only sensible across models employing the same response data
and with explanatory objectives with regard to available
data layers. So, these are the only model comparisons
we can provide.
Hierarchical Bayes models have a long and successful history by now in several areas but are just beginning
to be applied to problems in ecology (e.g., Borsuk et al.,
2004; Qian et al., 2003; Borsuk et al., 2001; Prato, 2000;
Clark, 2003). See Banerjee et al. (2003) for a general introduction to the methodology. However, to the best of
our knowledge, the methodology has not been applied
to study deforestation processes other than a preliminary paper we have published in the statistical literature (Agarwal et al., 2002). Before providing details
on our model development and specification in next
section, we would like to comment briefly on the dif-
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
ference between our approach and the simulation modeling approach that has been widely used to study deforestation (see, for example, Soares-Filho et al., 2002;
Duffy et al., 2001; Kok and Winograd, 2002). We remark the latter is really a different philosophic approach
to the problem compared to the explanatory approach
we adopt. While simulation-based modeling seeks to
predict observed land-use behavior by capturing the
known mechanistic aspects of the problem, our explanatory approach attempts to explain a response variable (in this case land use) in terms of other explanatory
variables. Moreover, simulation models are dynamic,
seeking to compare the evolution of deforestation. Our
model is static seeking to link land use to explanatory variables, in particular to population pressure. As
such, the two approaches are not directly comparable
to understanding deforestation patterns and processes.
In fact, the two approaches can in some cases complement each other—insights into the mechanistic aspects
of the problems can be incorporated when building
an explanatory model. However, this has rarely been
attempted.
3. Overview of model development and
specification
Our approach is to formulate a model for the joint
distribution of local human population attributes and
forest exploitation, given other explanatory variables
that are explicitly spatial. In so doing we are immediately faced with incompatibility of the data layers
(land use is obtained at the scale of 1 km × 1 km pixels (see below) while population counts are obtained
at irregular town-level polygons). We then overlay the
town-level map on the pixel-level map and modify town
boundaries, using majority pixel rule, so that each pixel
is contained in one and only one town. Upon such rasterization of the town data, we conduct the joint modeling at the pixel level. This joint distribution is specified by modeling the unobserved pixel-level population
counts and, then the conditional distribution of land use
given the associated count. The model for the latent
population counts leads to a model for the observed
counts. With two sets of spatial effects, one associated
with the town population counts, the other with pixel
level land-use model, a multilevel hierarchical model
results.
111
We show how such models can be fitted using Gibbs
sampling (Gelfand and Smith, 1990), thus allowing full
inference of all of the modeling levels (cf. Gelfand et
al., 2005). The misalignment problem extends recent
work of Mugglin and Carlin (1998), Mugglin et al.
(2000) and Gelfand et al. (2003). In the present context the other explanatory variables we employ are also
measured at the (l km × 1 km) pixel-level. However, if
this were not the case, a general strategy is as follows.
Identify the data layer at the finest spatial resolution.
For each remaining layer create an individual overlay
on this one. For each overlay, carry out a rasterization
so that each areal unit of the latter is contained on one
and only one unit of the former. Then model the joint
distribution of all the variable layers at this highest resolution. Such an approach yields a model for each data
layer at its observed scale. Note that it is not necessary to align all pairs of data layers. Rather, we need
only align each layer with the one at the finest spatial
resolution.
Two related implementation issues arise under our
modeling approach. First, for the response data we
work with, due to the presence of cloud cover, landuse classification is missing for approximately 8% of
the roughly 47,000 pixels. Thus, we introduce an imputation approach, i.e., a spatial multiple imputation
(Schafer, 1997), to handle this missing data. Second,
our modeling requires commitment of considerable
computational effort (tailored programs for the individual application) and computer run time. To justify
this commitment we offer model comparison revealing the inferential benefits of our modeling relative to
simpler model specifications, which can be fitted more
easily and quickly.
4. The dataset
The data used in developing the deforestation model
are from Madagascar, an area of the world designated
as particularly high priority for conservation efforts. It
is recognized as 1 of 25 mega-diversity countries in the
world (Meyers et al., 2000) and the eastern tropical wet
forest in Madagascar is globally one of 12 rain forest
“hot spots”. At the same time, the forests in Madagascar have been characterized as under tremendous
threat from deforestation (USDA Forest Service, 2000;
Sussman et al., 1994; Oxby, 1985). As a consequence
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many of the most well-known plant and animal species
are listed by the International Union for the Conservation of Nature (IUCN) as globally threatened.
There is considerable difference of opinion as to how
much of the original forest in Madagascar is left as we
pointed out earlier, but estimates range down to as little
as 10%, and perhaps only 25% of what is left is considered primary or undisturbed forest (Sussman et al.,
1994; USDA Forest Service, 2000). Of these systems,
the coastal forests are the most threatened. Estimated
deforestation rates for Madagascar are quite variable:
less than 1% per year to 5% or more. Some have estimated that within 20 years, little or no forest will remain
outside protected areas (currently about 1.85% of the
total land surface). But, as mentioned earlier, most of
these estimates of forest cover and deforestation rates
are speculative and controversial. Concerns over deforestation and attempts to estimate rates is nothing new.
In 1921 Perrier de la Bâthie estimated that between
1900 and 1920, 49,000 km2 of forest had been lost.
Early in the 19th century, the first King of Madagascar
proscribed the clearing and burning of forest (Verin and
Griveaud, 1968).
The focal area for this study is the wet tropical forest biome within Toamasina (or Tamatave) Province,
Madagascar. This province is located along the East
Coast of Madagascar, and encompasses the greatest extent of tropical rain forest on the island nation
(cf. Fig. 1). The aerial extent of Toamasina Province
is roughly 75,000 km2 . To model forest cover for
the province, we obtained or constructed five digital geospatial layers representing: town boundaries
(with asociated 1993 population census data), elevation, slope, road and transportation networks (1993)
and land cover.
4.1. Forest classification
A single NOAA-11 AVHRR, 10-day composite image for September 21, 1993 was obtained
from the USGS EROS Data Center (http://edcwww.
cr.usgs.gov). Three reflective bands (1, 2 and 3) plus
a Normalized Difference Vegetation Index (NDVI)
(bands 3 − 2\3 + 2), were presented to an unsupervised Iterative Self Organizing (ISODATA) clustering algorithm, within ERDAS Imagine v.8.4 software.
This clustering output 30 initial classes, which were
interpreted using expert personal knowledge of the
cover types, their composition, distribution and location within the region. Two Landsat 5 Thematic Mapper TM images (September 15, 1993 and November 21,
1994), a set of aerial photographs from the 1940s and
1960s together with extensive independently collected
ground-truthing data were used to aid fine scale pattern
delineation. The 30 classes were iteratively assessed,
reclassified with a supervised, maximum-likelihood
process and reduced to the final 5 used for the modeling:
mature forest, secondary patch forest, cleared forest,
non-forest and missing values (for small patch areas
obscured by clouds or cloud shadows, haze, etc.). We
believe this provides the most accurate characterization
of the landscape at this scale available. The vector maps
of forest cover currently available (Faramalala, 1995;
Du Puy and Moat, 1999) are overly smoothed and in
many cases report forested and non-forested areas that
based on our ground truth information, are misclassified.
4.2. DEM generation
The quality of existing global topographic information (Gtopo30, Digital Chart of the World) we found to
be inadequate, containing numerous artifacts and noise.
Instead we use vector elevation contour data (100 m)
digitized by the Madagascar geodetic survey office (Intitut National de Geodesie et Cartographie, or Foiben
Taosarin-tanin ’I Madagasikara (FTM)). These were
used to generate a continuous, 1 km × 1 km grid surface, utilizing a combination of Surfer v. 6.04 (Golden
Software, Golden CO) and Arc View Spatial Analyst v.
2.0 (ESRI, Redlands CA) software. Spatial misalignments and misregistration errors were corrected to produce the final 1 km2 elevation grid layer. In turn, this
layer was used to calculate a slope (degrees) layer, using Arc View Spatial Analyst v. 2.0.
4.3. Town, population and other cartographic data
Township (i.e., “firaisana”) boundary information
was digitized for all 162 firaisana in Taomasina
province by the Madagascar geodetic survey office
(FTM). Census data were obtained from the United
Nations via the Madagascar census bureau (Direction
et Statistique Sociales) for each firaisana for 1993.
Vector point data for various population center features (i.e., village/hamlet localities or “fokontany”),
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
firaisana “headquarter” villages, fivindronana headquarter towns, and the provincial capitol) were
digitized and supplied by FTM. Road and tracks were
also digitized and supplied by FTM (primary routes
(sealed roads) and railroads, motorable (in dry season)
tracks, rough ox cart tracks, and primary footpaths).
These vector layers were all rasterized to 1 km2 grids.
Grid cells fell along town boundaries were assigned
to a particular town by majority area rule. Grid cells
with one type of road were assigned the appropriate
value. When two or more road/path network types fell
in a particular grid cell, the more developed road value
was assigned to the pixel.
The final layers were rasterized to 1 km and clipped
to the extent of the Town boundaries in the province. All
islands were clipped as was the one town that covered
less than 1 km2 . All data were obtained in, or projected
to, Laborde (m), WGS84 coordinate values.
Following clipping, 159 townships and 74,607
1 km × 1 km pixels remained in our data set. Fig. 2
shows the town level map for the towns in the study
region. In the western portion of the province an escarpment demarcates the eastern wet forest or potential
forest biome from the western, seasonally dry grassland/savannas mosaic. Visual inspection of the data
layers prior to analyses revealed some differences between the northern and southern parts of the province.
The North had fewer population centers most of which
tend to be clustered near the coast, and larger forest
patches, while the South had more populations centers scattered across the landscape, with many smaller
forest patches and more extensive road development,
including primary routes to the national capital, west
of the study region. With concern that pattern and process might differ between these regions and with interest in making comparisons between them, we created two disjoint regions. In particular, we excluded
the western (non-forested majority) towns and introduced, between North and South, a buffer region at
least two towns wide (to provide separation of North
and South spatial effects). This yielded the North and
South regions identified in Fig. 2. The North included
46 towns encompassing 22,347 km2 pixels and total
population 707,786; in the South there are 66 towns
encompassing 24,623 such pixels and total population
664,066.
We note that in creating these two disjoint subregions, the concern was not to strive for homogene-
113
Fig. 2. Northern and southern regions defined within the study region. All town boundaries are shown. Population size categories are
shown for northern and southern towns studied.
ity of spatial pattern of the sub-regions. Rather, due
to the fairly simple mean specification of the model
described below, we were concerned that omitted or
unobserved variables which carry spatial information
may differentially affect land use in the sub-regions.
This implies that the spatial model for random effects
(which are introduced as surrogates for these variables)
may differ among sub-regions. Moreover, the regression coefficients of the observed explanatory variables
may also vary among sub-regions. In general, the message is that if, a priori, one suspects pattern or process is operating differentially in portions of the study
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Fig. 3. Histogram for population for the 159 towns in the full study area.
road classifications. In fact, because the sparseness patterns differed between the North and the South, we
used differently defined binary classifications for the
two regions. In the North, no roads was one of the binary road classifications while the other four road types
formed the second classification. In the South, no roads
and footpath formed one category while the other three
road types were lumped into a second category.
region, it is prudent to fit individual models to these
portions.
The GIS data layers used in the modeling are shown
in Figs. 2–7. In Fig. 2, town population data for
North versus South is overlaid on the town boundaries
for the entire province. Fig. 3 provides a histogram
of the population data for all towns included in the
model.
The ordinal land-use classifications for the northern and southern region are shown in Fig. 4: the most
degraded land is shown as non-forest (class 1), followed by cleared and degraded (potential) forest lands
(class 2), secondary patch forest (class 3) and mature
forest (class 4) and the missing value class. Relative
contribution (in percentage) of each class is provided
in Table 1. Fig. 5 provides grey scale maps for elevation
while Fig. 6 provides grey scale maps for slope.
Road classification was modified prior to full model
development. Initially, we had five road types assigned
to pixels, viz. no roads, footpath, rough track, motorable track and primary route or rail road (Fig. 7).
However, due to the sparseness of counts in some of
the categories, we needed to reduce this to a binary
5. Model details
We model the joint distribution of land use (L) and
population count (P) at the pixel level (1 km2 ). Let Lij
denote the land-use value for the jth pixel in the ith
township and let Pij denote the population count for
the jth pixel in
the ith township. The Lij are observed
but only Pi = j Pij are observed at the town level. We
collect the Lij and Pij into town level vectors Li and Pi
and overall vectors L and P. We also observe at each
pixel elevation, Eij , slope, Sij and road classification Rij .
Lastly, let δi denote a spatial random effect for township
i. We elaborate these effects below.
Table 1
Contribution of land-use classes in percentage
North
South
Class 1
(non-forest)
Class 2 (cleared/
degraded forest)
Class 3 (secondary
patch forest)
Class 4 (mature forest)
Missing
2.5
10.7
30.7
42.5
17.2
11.1
43.5
26.2
6.1
9.5
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
115
Fig. 4. Ordinal land-use classifications for the study regions.
In specifying the joint distribution, f(L, P|{Eij },
{Sij }, {Rij }, {δi }), we factor this joint distribution as
f (P|{Eij }, {Sij }, {Rij }, {δi })f (L|P, {Eij }, {Sij }, {Rij }).
(1)
We condition in this fashion since one of our objectives is to see if population provides a significant
effect for land use. Such explanation is evidently
not causal; we could equally well-condition P
on L.
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D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
Fig. 5. Grey scale elevation maps for the study regions displayed in a limited number of classes here.
Turning to the first distribution in (1), the population
model, we assume that the Pij are conditionally independent given the E’s, S’s, R’s and δ’s, i.e., we write
f (P|{Eij }, {Sij }, {Rij }, {δi })
=
f (Pij |Eij , Sij , Rij , δi ).
i
(2)
j
Moreover, since Pij is a population count and since
many of these (unobserved) counts will be sparse for
this region at this scale, we assume Pij : Poisson (λij )
where
log λij = β0 + β1 Eij + β2 Sij + β3 Rij + δi
(λi ) where log λi = log j λij = log j exp(β0 + β1 Eij +
β2 Sij + β3 Rij + δi ).
In (3), the β”s are regression coefficients while,
again, the δi ’s are township level spatial random effects. They are intended to capture anticipated spatial
similarity for neighboring townships with regard to
population. Unlike usual random effects which are
assumed to be independent and identically distributed
normal random variables, the distributions of the δi ’s
are specified conditionally given all of their neighbors.
That is,
(3)
Since the Pij ’s are conditionally independent Poisson variables, summing over j yields Pi : Poisson
δi |δj , j = i∼N
j ωij δj
j ωij
τ2
,
j ωij
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
117
Fig. 6. Grey scale slope maps for the study regions.
where ωij = 1 or 0 according to whether or not town
j is contiguous with town i and τ 2 is the spatial variance component. In other words, δi is expected to vary
about the average of its neighbors. Such models are referred to as conditionally autoregressive (CAR) models. See, for example, Besag (1974) or Cressie (1993)
for further details. We note that since the Pij are not observed we cannot introduce pixel level spatial effects
into the population model. We cannot adjust the mean
of population counts we have not seen; unstable model
fitting results. So, spatial adjustment to λij will be iden-
tical for all pixels within a town and will be similar for
pixels arising from adjacent towns. Also, since the Pij
are
conditionally independent random variables with
j Pij = Pi , we have {Pij }|Pi : multinomial(Pi ; {γ ij })
where γ ij = λij /λi .
In the second term in (1), the land-use classification
model, we assume conditional independence of the Lij
given the P’s, E’s, S’s and R’s. To handle the ordinal
nature of the Lij ’s, we follow Albert and Chib (1993),
introducing a latent variable Wij associated with each
Lij . Wij is conceived as a continuous random variable
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D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
Fig. 7. Vector road classification for the entire province.
on the real line, interpreted as the extent of forest
cover. We then imagine that the four ordinal land-use
classifications cut the real line into four intervals so
that Lij = l if Wij ∈ (γ l−1 , γ l ), l = 1, . . ., 4 with regard
to degradation where γ 0 = −∞, γ 4 = ∞. Hence,
πijl ≡ P(Lij = l) = P(Wij ∈ (γl−1 , γl )), l = 1, . . . , 4,
provides the four land-use classification probabilities
at each pixel in each town. In other words, Lij is a
multinomial trial with possible outcomes l = 1, . . ., 4
having respective probabilities πijl , l = 1, . . ., 4. The
Wij s can only be identified up to translations. That is, the
Wij ’s are located relativeto each other on this imaginary
scale but the data cannot absolutely place the Wij ’s. To
do so, without loss of generality, we can set the cut point
γ 1 = 0 but γ 2 and γ 3 are unknown. Hence, given Wij and
the γ’s, Lij is determined, i.e., the conditional distribution degenerates to a particular l given Wij . Conversely,
given Lij and the γ’s, Wij is restricted to an interval.
Thus, the land-use classification model becomes f(L,
W|{Eij }, {Sij }, {Rij }, {Pij }, γ 2 , γ 3 ) which we can
write as
f (L|W, γ2 , γ3 ) · f (W|{Eij }, {Sij }, {Rij }, {Pij }).
(4)
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
Again, due to the conditional independence, the first
distribution in (4) can be written as i j f(Lij /Wij |Eij ,
γ 2 , γ 3 ). The second becomes i j f(Wij |Eij , Sij , Rij ,
Pij ) where we use a linear regression model,
E(Wij ) = α0 + α1 Eij + α2 Sij + α3 Rij + α4 Pij .
(5)
However, note that because the Wij ’s are not observed,
their latent scale is not identifiable. We could multiply
each by a constant, multiply the α’s by this constant
and multiply the standard deviation of Wij by this constant and the modeling would be unaffected. Hence,
without loss of generality, we set var(Wij ) = 1. Note
further that we cannot introduce spatial random effects
(in fact any random effects at the pixel level) into the
land-use model. Were we to propose including effects
say ϕij in (5), since Pij is not observed, the data would
not be able to separate Pij and ϕij in the component
α4 Pij + ϕij . Nonetheless, the δi ’s in (3) induce spatial
smoothing in explaining the Pij ’s which, in turn,
produces some spatial smoothness in explaining the
Wij , hence Lij . A simpler version of the model in
(1)–(5) is presented with some analysis along with a
more technical perspective in Agarwal et al. (2002).
We conclude with several remarks:
Remark 1. A more customary modeling approach to
explain the Lij would be a non-hierarchical categorical
π
,
regression. For instance, we could model logit πij,l−1
ij,l
l = 2, 3, 4 as a linear model in Eij , Sij and Rij . Such
models can be routinely fitted using standard statistical software packages but since we do not have Pij ,
what should we do? Set Pij = Pi ? Set Pij = Pi (area of
ijth pixel)/(area of township i)? These spcifications are
equivalent since all pixels are the same size. However,
the assumption of a uniform distribution across townships is inappropriate as Fig. 9 below reveals. By introducing the latent Pij ’s, the Eij , Sij , Rij , and δi allow
us to learn about them while reflecting our uncertainty
in their actual values.
Remark 2. More generally, note how our modeling
explicitly handles the misalignment issue. To build a
model at the town level for the Pi , what should we use
for Ei , Si , and Ri ? To build a model for Lij , we should
introduce a Pij .
Remark 3. It is perfectly fine to use the covariates
Eij , Sij , and Rij , to characterize the joint distribution
119
of Lij and Pij . However, with regard to conditional and
marginal specifications, if we use these covariates to
explain Pij , should we also use them to explain Lij
given Pij (or perhaps, vice versa)? In familiar normal
linear modeling, we would introduce redundant
parameters in the marginal model for Lij . The marginal
model is over-parametrized, the regression coefficients
are not identified. Within the Bayesian framework, no
problem arises. We would find little learning about the
components of the coefficient vector but no problem
with the overall coefficient. In the present situation,
with a Poisson model for the Pij ’s, the identifiability
problem does not arise; in the marginal model for Lij ,
we have both an additive form and a multiplicative
form in the covariates. Perhaps more importantly, the
marginal model for Lij would not involve Pij . We are
explicitly interested in a conditional explanation for
Lij given Pij and the other covariates.
6. Fitting and inference for the model
The model defined by (1)–(5) is referred to as a multilevel or hierarchical specification. In fact, at the lowest
level we have the observed land-use classifications. At
the level above we have the latent W’s. At the top level
we have the latent pixel level populations constrained
by the observed township populations. This model is of
high dimension with unknowns, {Wij }, {Pij }, {δi }, α,
β, γ 2 , γ 3 and τ 2 . Fitting of this entire model is feasible
only within a Bayesian framework. That is, we have
already provided the joint distribution for L, W and P
given the model parameters. If we view these latter
unknowns as random and add a so-called prior distribution for them we have a complete model specification, i.e., a specification for the joint distribution of all
the variables in the hierarchical model. With this joint
distribution we can provide inference regarding any aspect of the model. The prior distribution for the δ’s was
described above. For α, β, γ 2 , γ 3 and τ 2 , the general approach is to use whatever prior or partially data-based
information we may have regarding these unknowns to
obtain some idea of what range they are likely to fall in.
We use this range to develop proper prior distributions,
which are as non-informative (i.e., as vague) as possible (in order to let the observed data drive the inference)
while still retaining stable computation. Details of the
prior specifications are supplied in Appendix A.
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The resulting model is fitted using simulation-based
methods, i.e., Gibbs sampling (Gelfand and Smith,
1990) and Markov chain Monte-Carlo methods (see,
for example, Gilks et al., 1996). The output from such
simulation is sampled from the joint posterior distribution of all model unknowns. Unfortunately, such model
fitting requires considerable effort and time. (In fact,
we summarize the required full conditional distributions and provide some pseudo-code for implementing the sampling in Appendix B.) The reward is exact
inference (without relying on possibly inappropriate
asymptotics) and more accurate measurement of variability by capturing the uncertainty regarding the model
unknowns, which is not obtained using classical statistical approaches. In fact, we know of no other way to
fit models (1)–(5). Simplified versions may be fitted
using the E–M algorithm (Dempster et al., 1977) to
obtain likelihood-based inference. But then we have
concerns regarding associated asymptotic variance estimates. It may also be possible to fit the model stages
separately rather than simultaneously. Again, we are
concerned that this will misrepresent the extent of variability. These relate to issues of model choice, which
we discuss below.
7. Imputing missing land-use classification
Table 1 shows the distribution of land-use classifications for the North and South regions. Note that
6.1% of classifications are missing (primarily due to
cloud cover) in the North and 9.5% are in the South.
One of our fundamental modeling assumptions is that
each pixel belongs to one and only one town and the
pixel level population, when aggregated over a town,
gives the known town level population. Thus, we cannot
delete pixels with missing land-use classification from
our analysis. Rather than relying on subjective or ad hoc
determination of these missing pixel values, we elect to
impute missing land-use values in order to have a complete set for fitting our hierarchical model. In fact, the
imputation provides an entire set of missing land-use
classifications using a neighbor-based joint spatial distribution for the pixel values. Such a distribution should
support (but not require) land use for a given pixel to
be similar to that of its neighbors. Moreover, we do
several such imputations to see the sensitivity of our
resulting inference to the imputation, to better assess
the variability in the model unknowns. Various imputation models could be adopted; we employed a convenient Potts model (Potts, 1952; Green and Richardson,
2002). Details are supplied in Appendix C. We used
three imputations, finding negligible differences in the
resulting parameter estimates across these imputations.
8. Model determination
The model formalized in (1)–(5) is elaborate and
challenging to fit. Is the effort justified? Here we consider three simplifications resulting in easier to fit models and, using a model choice criterion, show that they
are not as good.
For instance, a crude model could ignore the population model (3). Instead, we could fit the land-use
model in (5) inserting for the unknown Pij ’s an areally
allocated portion of the total township population to
each pixel within the township. In other words, if there
are ni pixels in township i, we set Pij = Pi /ni . Hence we
have only the one stage model in (4) and (5) to fit.
An improvement in this naive approach first fits the
model in (3) (with or without spatial effects). This fitting considers the Pi ’s as the only data, the Pij ’s are
latent variables, and estimates the Pij ’s along with β
(and δ if spatial effects are included). The resultant fitting provides posterior means E(Pij |Pi ) which could
then be inserted into the land-use model (5). These expected values are anticipated to provide better estimates
of the true Pij than those from the crude allocation. Furthermore, with the inclusion of the δi , the E(Pij |{Pi })
will tend to be smoother than without. But regardless,
we are still failing to capture the uncertainty in not
knowing Pij ; we will underestimate the variability in
the overall model. We note that in the absence of spatial effects, our fitting approach is analogous to the use
of the E–M algorithm to obtain maximum likelihood
estimates (Dempster et al., 1977) with incomplete data
(the Pij are missing). In fact, we can also obtain the customary likelihood inference using the E–M algorithm.
Again, concern is with the resultant interval estimates,
i.e., the approximate normality assumption on which
they are based and the goodness of the variability estimates (estimated standard errors) which they require.
To compare the foregoing models we use a version
of the posterior predictive loss approach in Gelfand
and Ghosh (1998). Using squared error loss for con-
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
121
Table 2
Model comparison for the North and South regions
North
South
Model
G
P
D
G
P
D
I (without population)
II (without spatial)
III (full bayesian)
10949.68
10776.78
10751.74
11911.42
11489.25
11460.42
22861.10
22266.03
22212.16
15317.60
14667.41
14670.59
16788.42
15407.17
15404.39
32106.02
30074.58
30074.98
Gi : goodness of fit term; Pi : penalty term; Di : combined.
venience this approach provides a criterion, which is
composed of two parts. One measures goodness of fit
(G) of the model; the other is a penalty term (P), penalizing model complexity. In other words, more complex
models will tend to fit better but should be penalized
for their “size” in order to encourage parsimony.
Specifically,
G=
ni 4
159 (Lijl,obs − πijl )2
i=1 j=1 l=1
and
P=
ni 4
159 V (Lijl )
imputation was implemented apart from the model
fitting, we chose the latter.
It is useful to record both G and P (in fact, in some
cases it might be useful to plot the components of G
and P. However, if we seek to reduce model choice to
a single number, then we choose the model which minimizes D = G + P. Table 2 presents the values of D, G
and P for both the North region and the South region for
Model I (simple allocation, without population model)
Model II (improved model, includes population model,
but not spatial effects) and Model III (the full Bayesian
model in (1)–(5)). Clearly, Models II and III are preferred to Model I and, at least in the North, Model III
is preferred to Model II.
i=1 j=1 l=1
where Lijl,obs = 1 if Lij,obs = l, 0 otherwise,
πijl = P(Lij = l|Lobs ) and V(Lijl ) is the predictive
variability of Lij , i.e., πijl (1 − πijl ). Recall that for
each i, j, one Lijl,obs = 1 and the remaining three are
0. Under G, a model is incrementally rewarded at
an i, j pair if its πijl ’s (which also sum to 1) behave
similarly to the Lijl,obs . Under P, a model is rewarded
at i, j if it produces “informative,” i.e., extreme πijl ’s
. In calculating G and P we could sum over only the
observed Lij ’s or also over the imputed Lij ’s. Since the
9. Model results
Table 3 provides point estimates and 95% credible
intervals for both the population and land-use models. For comparison, Table 4 presents point estimates
and associated interval estimates for Model II fitted using an E–M algorithm. We see that with regard to the
population model, the two models provide very similar
point estimates but that the approximate likelihood in-
Table 3
Median and 95% credible interval (in parentheses) for the full Bayesian Model III
Population model parameters
β1 (elevation) × 10−4
β2 (slope)
β3 (roads)
τ δ2
Land-use model parameters
α1 (elevation) × 10−3
α2 (slope)
α3 (roads)
α4 (population)
North
South
−8.210 (−8.412, −8.038)
.0704 (.0682, .0725)
1.157 (1.151, 1.163)
1.503 (1.058, 2.423)
−6.487 (−7.028, −5.986)
−.0226 (−.0302, −.0150)
.2124 (.1922, .2314)
1.791 (1.297, 2.455)
4.144 (4.024, 4.242)
−.0089 (−.0198, .0020)
−.5931 (−.6649, −.5232)
−.0101 (−.0119,−.0081)
1.942 (1.874, 2.018)
.0415 (.0306, .0526)
−.4587 (−.5373, −.3892)
−.0028 (−.0033, −.0024)
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Table 4
Point estimates and 95% interval estimates (in parentheses) obtained from the EM algorithm (Model II)
Region
North
South
Population model parameters
βl (elevation) × 10−4
β2 (slope)
β3 (roads)
−8.217 (−8.287, −8.147)
.0703 (.0695, −0711)
1.157 (1.154, 1.160)
−6.397 (−6.467, −6.327)
−.0241 (−.0253, −.0229)
.2182 (.2151, .2213)
Land-use model parameters
α1 (elevation) × 10−3
α2 (slope)
α3 (roads)
α4 (population)
4.304 (4.292, 4.316)
−.0075 (−.0191, .0041)
−.7377 (−.8232, −.6522)
−.0019 (−.0033, −.0005)
ference in Model II produces much tighter confidence
intervals revealing the significant underestimation of
uncertainty. For the land-use model, at least for elevation, this continues to be the case. Thus, for the remainder of this section we describe the results for Model III.
For this model, lower elevation and presence of road
networks are associated with population settlement in
the North as well as the South with the effects being
more significant in the North. In the South, lower slope
is associated with higher expected population, which is
consistent with what we anticipate. In the North, however, higher slope is associated with higher expected
population, contrary to what one might expect. Turning
to the land-use model, the coefficient for population although small is significantly negative for the North and
the South with the influence being more pronounced in
the North. In fact, the interval estimates for the North
and South do not overlap. Increased population pressure is associated with increased chances of deforestation with the effect being less in the South. Elevation
is associated with forest cover with the effect being
more pronounced in the North. Here too, the interval
estimates for the two regions don’t overlap. Presence
of road networks significantly increase the chance of
deforestation in the North as well as South. Slope is
mixed with regard to significance. While it emerges as
insignificant in the North, it significantly increases the
chances of forest cover in the South.
Recall that the spatial random effects, the δi ’s, were
introduced to provide spatial smoothing to the population model which in turn induces such smoothing to
the land-use model. A priori, under the CAR model,
all δi ’s have essentially the same distribution and mean
0. A posteriori, spatial pattern emerges. To see this,
in Fig. 8 we present a map of the posterior means of
2.084 (2.076, 2.092)
.0621 (.0499, −0742)
−.4744 (−.5548, −.3940)
−.0024 (−.0028, −.0020)
the δi ’s.The figure clearly reveals that spatial proximity encourages similarity in the expected spatial effects
of population processes. Here, lighter colors indicate
a negative expected spatial effect, i.e., the adjustment
to the mean population for pixels in that township will
tend to diminish population, vice versa for the darker
colors.
We have claimed that a model for pixel population
such as (3) is more effective than a naive areal allocation of township population to pixels. The results of
the model comparison presented in Table 2 certainly
support this. Here, we present some further informal
support. In Fig. 9 we have created a map of the posterior mean populations (on the square root scale) at
the pixel level. The square root scale is employed since
this is the customary symmetrizing and variance stabilizing transformation for count data. The smoothing
of the population is evident at finer pixel resolutions.
Overlaid on this map is the population center dot map.
The dots locate the villages, town centers or larger administrative centers. The size of the dot roughly corresponds to population size. Though the dot map could
not be reliably used to model pixel level population,
it is evident from Fig. 9 that, generally, there is good
agreement between the maps.
10. Discussion
Returning to the model issues section, we summarize first what has been accomplished. We have
formulated an ordinal land-use classification scale.
We have built a phenomenological hierarchical model
at the pixel level, which explains land use given
population and other covariates as well as population
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
123
Fig. 8. Population model spatial random effects shown for the study regions.
given other covariates. We have accommodated the
misalignment between the population data layer (at the
township level) and the land-use data layer (on a 1 km2
pixel grid). We have employed a spatial imputation
(simulation) strategy to handle the missing land-use
pixels in an objective fashion. We have introduced
spatial smoothing for the population model, the
only feasible possibility given the data available. We
have uncovered significant explanation in the mean
structure from the elevation, slope and roads data
layers in both the population and land-use models.
In this regard, we have found a significant negative
coefficient for population in the land-use model.
Increased population pressure increases the chance
of forest degradation. We have argued, through a
model comparison criterion, that a more sophisticated,
hierarchical model, which includes spatial effects
explicitly and a population model outperforms models
lacking these specifications. We have also argued and
demonstrated that our hierarchical model captures
uncertainty, which was underestimated by simpler
models. Lastly, we have shown the sort of statistical
inference that is possible with our hierarchical model,
particularly with regard to the spatial smoothing.
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Fig. 9. Expected population for the study regions (on the square root scale) at the pixel level with the population vector point map overlaid.
10.1. Historical context of landuse patterns in
Madagascar
Before we discuss the patterns revealed in our analysis, it is instructive to review the historical context of
land-use change in eastern Madagascar. Based on our
earlier summary of deforestation trends, controversy
surrounds issues of rates and timing of forest loss in
Madagascar. Claims that most of the forest loss has occurred in recent decades are clearly misleading. The
French colonial archives clearly shows concern over
forest loss throughout French occupancy (Jarosz, 1993;
Perrier de la Bâthie, 1921). In fact the claim, certainly
exaggerated, was made that some 70% of the primary
forest was destroyed between 1895 and 1925 (Hornac,
1943–1944, cited by Jarosz, 1993). In any case the
standard explanations offered for forest loss was the
extensive nature of traditional slash-and-burn upland
rice culture practiced by the locals, forest concessions
encouraged by the French with associated destructive
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
logging practices, and the advocacy of selective forest
conversion to plantation and cash crops (Jarosz, 1993;
Perrier de la Bâthie, 1921). It is likely that all of these
activities would have been encouraged by any site factors that eased access to forested blocks, i.e., adjacency
to villages and towns, access by road or even footpath, and lower, shallower slopes. Land tenure systems
changed with the 1926 decree that all vacant land be declared state land. Local populations no longer held title
to ownership over ancestral lands that might currently
be unoccupied (Esoavelomandroso, 1985). Moreover,
during the late 19th and early 20th century several laws
were passed prohibiting the burning of forests (Jarosz,
1993). Together these laws may well have acted to disenfranchise local populations with respect to traditional
land conservation practices, actually exacerbating land
degradation.
Yet despite these observations, there are many indications that deforestation processes have been on going for centuries. Few scholars have looked for older
records of forest extent in Madagascar. Such exist,
in the form of historical accounts of the 17th–19th
century (Ellis, 1858; Flacourt, 1658; Grandidier et
al., 1920). These narratives make it clear that the
distribution of forests were patchy and geographically patterned long before the French colonial period. It is likely that modern patterns of forest distribution reflect, in part, forest distributions of the relatively distant past. The patchy nature of the eastern wet forests was evident to mid-17th century explorers (Grandidier et al., 1920, cf. Flacourt, 1658:
Map of Isle Ste. Marie and adjacent Madagascar mainland), as well as the early and mid-19th century travelers (Hébert, 1980; Ellis, 1858). All of these early
accounts describe large tracts of deforested landscape
from the sea up to 600 m on the mountains. This
landscape was clearly a patchwork of cultivated and
grazed lands, secondary forests, abandoned lands, and
mature forests. Ellis (1858) characterized the limit
of closed forest in the mid-1800s as a band along
the eastern mountains from 25 to 40 miles wide, not
much different from what is found today. There is
some evidence that extraction of forest resources was
frequent and widespread, together with land clearing
along the east coast during the 7th–11th century era
(Domenichini-Ramiaramanana, 1988; Dewar, 1997)
continuing up to the beginning of the colonial era
(Kent, 1992).
125
The early accounts provide a picture of periodic cycles of landscape clearing for agriculture and abandonment with some evidence for forest regrowth. In part
this reflected local cycles of human population growth
and decline (Campbell, 1991; Kent, 1992), with the result being a very heterogeneous landscape. The historical perspective is much clearer for the northern section
of the region than the southern. This reflects the infrequent visits of Europeans to the southern half of our
east coast focal region.
In the central northern area (around Fenerive), there
was a shift in human settlements evident in about the
17th century (Wright and Dewar, 2000), with the abandonment of coastal villages, and the establishment of
new villages on defensible hilltops at a distance from
the coast (cf. Flacourt, 1658: Map of Isle Ste. Marie
and adjacent Madagascar mainland). This is also reflected in the narratives of 19th century travelers (cf.
Grandidier et al., 1920; Ellis, 1858). Many of these
new villages were much larger than previous occupations, so that local population densities were increasing, and may have led to concomitant increase in local
anthropogenic effects on the countryside. All of these
changes are most plausibly accounted for by an increase in insecurity, and the virtual absence of trading
or slaving along the southern coastal area (Kent, 1992),
would have meant that population distributions and
local environmental effects likely followed divergent
paths.
10.2. Forest changes over the past century
Putting the historical information into the context
of current land use, it is clear that deforestation has
continued up to the present, exacerbated by population growth. The trend has tended to reduce the number and extent of patches of secondary and mature
forests in the landscape. This is easily seen on comparing aerial photographs from the 1940s and 1960s,
detailed forest patches mapped in the 1960s by Dandoy
(1973), and by Benoit de Coignac et al. (1973) about
the same time, with recent ground truth and satellite images (data not shown). The result today is a
much more homogeneous landscape of degraded lands,
with any remaining patches of forest relegated to hill
crests and protected valleys. Only along the highest
mountain slopes does one find continuous tracts of
closed, mature forest. These have retreated in recent
126
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
decades, but maps of large forest tracts today are surprisingly similar to those of 1969, 1934, 1921 and
1902.
The maps provided by Humbert et al. (1964–1965)
have figured prominently in depicting current forest cover for our region (Du Puy and Moat, 1999)
and in estimating deforestation rates (Green and
Sussman, 1990). Original forest cover was taken from
the thumbnail “Types de Végétation” maps, which purport to show potential vegetation in the absence of
man (Humbert and Cours Darne, 1965; pp. 152–154).
These are conjectured maps with little supporting information, and are primarily constructed from the main
“Végétation Naturelle” maps of the east coast. These
are used as the baseline “original extent” of forest cover
by Green and Sussman (1990), but these provide a misleading benchmark at best. The main “Végétation Naturelle” maps of Humbert and Cours Darne are used
to portray current forest cover in the most recent maps
of forest cover (Faramalala, 1995; Du Puy and Moat,
1999), as well as forest cover in 1950 (Green and
Sussman, 1990). The Humbert and Cours Darne maps
were assembled by committee and the forest classification was based on surrogate elevation, bioclimate
and edaphic variables with uncertain use of aerial photography, and little or no ground truthing. From our
own experiences with aerial photography of the region,
vegetation classification without accompanied ground
truthing is difficult and misleading. Based on all available information it is apparent that the forest cover
maps produced by Humbert and Cours Darne provide
a gross over estimate of the extent of forest either currently or for any point over the past half century. Certainly this is true based on even a cursory comparison of
the Cartes Forestier with current or recent forest cover
on the ground.
10.3. Demographic trends
One can gain some insight on past populations patterns across the landscape from the available demographic maps (de Martonne, 1911; Gourou, 1945; Le
Bourdiec et al., 1969). All show a trend North to South
that is reflected in the population densities observed in
1993 (cf. Fig. 8). In the South the populations over the
past 90 years have tended to be more evenly spread
across the landscape. In the North the population centers tend to be more restricted to areas nearer the coast.
Over that time, population densities have simply increased across the region.
10.4. Interpretation
How does this information integrate with the results
of our spatial analysis of land-use patterns? Population
levels are negatively associated with elevation and
forested landscape are positively associated with elevation in both the North and the South. This obviously
reflects the current and historical location of settlement
area at low to moderate elevation and the fact that most
of the remaining forest blocks are spatially restricted
to the highest mountain slope or locally on hill tops.
The effect of elevation is much higher in the North
than South. This reflects undoubtedly that fact that
large blocks of forest are associated with the higher
mountains of this section of the province. Slope is negatively associated with populated blocks in the South
but positively associated in the North. This may reflect
the fact that settlements were shifted to fortified hill
tops (but not at great elevation) in the North beginning
in the 17th century (cf. Flacourt, 1658: Map of Isle Ste.
Marie and adjacent Madagascar mainland—the center
of our North region; no comparable maps are extant
for the South). Pixels with hill tops would tend to
reflect higher slopes. However, this trend slope was not
observed in the South. Rather settlements tend to be
spread out more evenly across the landscape over time
in areas of convenience, likely avoiding areas of slope
and elevation. Slope shows a similar trend in the spatial
distribution of forested blocks, North versus South. In
the South steep slopes are positively associated with
forested blocks. In the North there is no trend with
slope. This may reflect the trend of finding some hill
tops (and hence steep slopes) occupied by towns and
some by forests. An alternative explanation is that
the North may represent a rougher topography with
greater elevational change but also with greater slope
change within and among pixels. The net result may be
little effective signal of slope on most of the landscape
occupied by forest. We explore this possibility next.
While there is no generally accepted measure of spatial
roughness for an areal unit, a measure of local variation
proposed by Philip and Watson (1986) is widely used.
This measure is computed through automatic triangulation at locations within the areal unit. Larger values
of this index indicate greater roughness. In Table 5, we
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
Table 5
Roughness comparison between North and South regions
Quantiles
Region
North (21466)
South (22950)
.025
0
0
.25
.0002
0
.5
.0033
.0018
.75
.0072
.0061
.975
.0403
.0369
summarize the distribution of this index over 21,466
pixels in the North region and 22,950 pixels in the
South region. The table offers descriptive comparison
since there is no stochastic modeling of this index.
However, the quantiles for the distribution of roughness are somewhat larger in the North than the South
providing quantitative support for a somewhat rougher
topography.
11. Conclusions
To summarize, we have provided a new framework
to explain deforestation patterns in the presence of misalignment of data layers, missing data arising from
cloud cover, and incorporating spatially explicit structure through the use of a bayesian hierarchical model.
Although illustrated in the context of Madagascar, the
methods are more generally applicable to other ecological processes as we have noted at the end of Sections 1 and 2. Returning to an examination of population block effects, these are negatively associated
spatially with forested blocks, although the effect is
fairly weak. This weak signal may reflect the pervasive influence of slash-and-burn agriculture practiced
throughout the landscape even areas with low population densities. This agricultural practice may be more
determined by accessibility than proximity to population centers (Oxby, 1985). Elevation obviously plays
into this, as do roads of any sort, or indeed even footpaths. These transportation networks are negatively associated spatially with forest blocks and positively associated with inhabited blocks. Many of these routes
of transport have been used for centuries. In fact the
paths followed by Ellis in the mid-1800s continue as
access routes used today. In sum elevation along with
road/path networks, and to a lesser extent population
patterns play an important role in explaining the spatial distribution of forested blocks in the landscape.
But in addition historical patterns of land use continue
to play a pervasive role in the distribution of popu-
127
lation centers and forested patches observed today in
the landscape and appear to account for the differences
seen between the northern and southern sections of the
province.
Generalizing from our specific results, it is evident
that the structural hierarchical modeling style we have
employed is applicable to the modeling of land-use patterns in many other contexts. Similarity in land use between neighboring areal units would be expected, and
spatial random effects could be used to capture such association. Misalignment problems among data layers,
hence, between response and explanatory variables are
also likely to be common themes.
Our present application has led to a model for land
use, which is static, since we lack temporal information. However, were data available across time, we
could extend the modeling in (1)–(4). Formally, we
need only add a subscript t to those measurements
which change over time. Mechanistically, we might
think of the land-use process as evolving in both space
and time. Spatio-temporal random effects, δit could be
introduced to capture association across both space and
time.
Lastly, while the present setting has an ordinal categorical response variable, in other applications the response could be binary, e.g., presence or absence of a
species, or a count, e.g., abundance of a species. The
first stage model for the response would change to reflect this but hierarchical modeling with spatial structure could still be employed and would provide the
same benefit in terms of richer inference than is available with standard methods.
Acknowledgements
This project was supported by grants from NSF
(DMS 99-71206) to A.E. Gelfand and from the John
T. and Catherine D. MacArthur Foundation to R.E.
Dewar and J.A. Silan-der, Jr. Roland de Gouvenain,
Steven I. Citron-Pousty, Daniel L. Civco kindly provided assistance on some of the GIS component analyses of this study. Thomas H. Meyer provided feedback and discussion on spatial indices of roughness,
and implemented software for estimating the index
we use here. Andrew Latimer, Robin Chazdon and
Richard Primack provided critical comments on the
manuscript.
128
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
Detail on prior specifications
To complete the specification of the hierarchical
model we require priors for α, γ, β, γ 2 , γ 3 and τ 2 . To
obtain a well-behaved chain, we use priors for α and
β that are “data-centered” with large variances. Inference is not sensitive to this centering (we could use say
mean 0 centering) but our Markov chain Monte-Carlo
algorithm burns in more rapidly with such centering. In
particular, we assume β∼N(β̂, kDβ ) where β̂ is a point
estimate obtained from an expectation maximization
algorithm (Dempster et al., 1977), which imputes population at the pixel level and Dβ is the corresponding
dispersion matrix. For α, we adopt N(α̂, kDα ) obtaining the estimates by fitting a multinomial ordinal response model (using standard statistical packages like
SAS) that areally allocates the town population to each
pixel within the town. We experimented with several
values of k and found little sensitivity for k > 10. For
the two unknown cut-points γ 2 and γ 3 , we assume a
uniform prior subject to the constraint 0 < γ 2 < γ 3 < ∞.
For τ 2 we adopt an inverse gamma prior. In particular
τ 2 ∼ IG(2, .23). This specification has infinite variance
with mean roughly the sample variability in the log λ̂i
(where λ̂i = Pi. ). As is customary, to ensure identifiability, i.e., a well-behaved
posterior distribution, we
impose the constraints i δi = 0. (Besag et al., 1995).
Brief detail on model fitting
We briefly describe the conditional distributions of
each variable and how they get updated (simulated) in
the gibbs sampler. At time t we denote this operation by
gibbsup-date (θ t , θ t+1 ) where θ is a vector of all model
unknowns that are to be updated.
• We assume the W’s have a normal distribution
with mean given by (5) and variance 1. Then
Wij ∼N(E(Wij ), 1)1(γLij −1 < Wij ≤ γLij ) and gets
updated using draws from univariate truncated normals.
• γ l ∼ Uniform(a, b); a = max(␥l−1 , max(Wij s.t.
Lij = l)); b = min(γ l+1 , min(Wij s.t. Lij = l + 1)), l = 2,
3, . . .
−1
• α∼N((X X + Dα−1 /k) (X W + Dα−1 α̂/k),
−1
(X X + Dα−1 /k) ) where X is the design matrix that results from the linear regression in
(5). Sampling from a multivariate distribution is
standard.
• Pi ∼ multinomial(Pi ,λij /λi )) j exp(−.5(eij −α4 Pij )2 )
where eij = Wij − α0 − α1 Eij − α2 Sij − α3 Rij , i = 1,
. . ., T where T is the number of townships. We
update Pi using a Metropolis step with proposal
density which is multinomial (Pi , λ∗ij /λ∗i. ) where
λ∗ij = λij exp(α4 eij ). We have found this to work
very well with acceptance rate of around 50%.
• δi ∝ exp(−λi )λPi i N( j ωij δj / j ωij , τ 2 / j ωij ),
i = 1, . . . , T . This is log concave and updated
using adaptive rejection sampling (Gilks and Wild,
1992). Code is available from http://www.mrcbsu.cam.ac.uk/BSUsite/Research/ars.shtml
P
• βl ∝ ( i j exp(−λij )λijij )N(β̂, kDβ ), l = 0, 1, 2, 3.
This is also log concave and up dated using adaptive
rejection sampling. Take care with regard to underflows when evaluating the log density.
2
gamma (shape = T/2+2, scale = . 23 +
τ ∼ Inverse
2
i–j (δi −δj ) /2) where the relation i–j means i and j
are neighbors. This is easily sampled using standard
routines for simulating from the gamma distribution
(see Devroye, 1986).
Psuedo code to fit model HI
• Initialize θ to θ 0 . We set Pij0 = Pi /ni and taking α0 ,
γ 20 , ␥30 estimates obtained from fitting a multinomial ordinal response model and β0 estimates obtained by fitting Poisson regression assuming P = P0 .
The other parameters would be automatically initialized by the operation gibbsupdate(θ 0 ,θ 1 ).
• for i in 1:(B + Th × Nsamp) gibbsupdate(θ i−1 , θ i )
• Discard the first B iterates, store every Thth iterate
thereafter and use the Nsamp iterates to make inference
For the current application, B = 20000, Th = 50
and Nsamp = 1000 were appropriate.
Details on the multiple imputation
The imputation for all of the pixels with missing
land-use classification is carried out in an iterative fashion using a joint spatial Potts distribution. Recall that
Lij denotes the classification for the jth pixel in the ith
town and L denotes the set of all Lij . The joint density
D.K. Agarwal et al. / Ecological Modelling 185 (2005) 105–131
of L under the Potts model is
f (L) ∝ exp(φ > U(L))
129
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where U(L) = &1(Lij = Li j ) and the summation is over
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association while φ > 0 encourages classification agreement between adjacent pixels. Hence the conditional
distribution of Lij given all of the other pixels
f (Lij |remaining L s) ∝ exp(φ
1(Lij = Li j )
(C.2)
where the summation is over the neighbors of Lij . In
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